An important point is that the critical coupling exists only in asymptotic AdS\(_3\)  spacetime; it does not exist in asymptotically flat spacetime (\(\Lambda _{eff}=0\) ) where the VEV is a fixed non-zero constant independent of \(\xi \) . However, the non-minimal coupling term still plays a significant role in a flat background. In \(2+1\) -dimensional General Relativity without cosmological constant, it is well known that outside matter the spacetime is locally flat but has the topology of a cone whose deficit angle is proportional to the mass [1]}. However, we found that the deficit angle was not determined solely by the mass of the vortex but also depended on the coupling \(\xi \) . One remarkable consequence of this is that a higher mass did not necessarily yield a higher deficit angle.
